L^2 and intersection cohomologies for the reductive representation of the fundamental groups of quasiprojective manifolds with unipotent local monodromy

Let X be a projective manifold, and D be a normal crossing divisor of X. By Jost-Zuo's theorem that if we have a reductive representation \rho of the fundamental group \pi_{1}(X^{*}) with unipotent local monodromy, where X^*=X-D, then there exists a tame pluriharmonic metric h on the flat bundle \mathcal V associated to the local system \mathbb V obtain from \rho over X^*. Therefore, we get a harmonic bundle (E, \theta, h), where \theta is the Higgs field, i.e. a holomorphic section of End(E)\otimes\Omega^{1,0}_{X^*} satisfying \theta^2=0.
In this paper, we study the harmonic bundle (E,\theta,h) over X^*. We are going to prove that the intersection cohomology IH^{k}(X; \mathbb V) is isomorphic to the L^{2}-cohomology H^{k}(X, (\mathcal A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D)).